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U.U.D.M. Project Report 2016:45

Examensarbete i matematik, 15 hp

Examinator: Veronica Crispin Quinonez

December 2016

Department of Mathematics

Uppsala University

Asymptotic Expansions of Integrals and the

Method of Steepest Descent

Erik Falck

Asymptotic Expansions of Integrals and the

Method of Steepest Descent

Erik Falck

December 16, 2016

Abstract

This paper gives an introduction to some of the most well-known meth- ods used for nding the asymptotic expansion of integrals. We start by dening asymptotic sequences and asymptotic expansion. The classical result Watson's lemma is discussed and a proof of Laplace's method is presented. The theory of the method of steepest descent, one of the most widely used techniques in asymptotic analysis is studied. This method is then applied to calculate the asymptotics of the Airy function and the linearized KdV equation. 1

Contents

1 Introduction 3

2 Elementary asymptotics and Watson's lemma 4

2.1 Asymptotic expansion . . . . . . . . . . . . . . . . . . . . . . . .

4

2.2 Watson's lemma . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

3 Laplace's Method 10

4 Method of Steepest Descent 13

4.1 The idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

4.2 Steepest descent paths and saddle points . . . . . . . . . . . . . .

13

4.3 Leading order behavior ofF() . . . . . . . . . . . . . . . . . . .16

5 Applications 18

5.1 Airy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

5.1.1 Asymptotics asx!+1. . . . . . . . . . . . . . . . . .20

5.1.2 Asymptotics asx! 1. . . . . . . . . . . . . . . . . .24

5.2 The KdV equation . . . . . . . . . . . . . . . . . . . . . . . . . .

26

5.3 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . .

30

6 Acknowledgments 30

7 References 31

2

1In troduction

Many problems in mathematics and physics have solution formulas that can be represented as integrals depending on a parameter, and one is often interested in the behavior of the solution as the parametergoes to innity. The most frequently used way of evaluating how these integral representations behave is to nd an asymptotic expansion in the parameter. In many problems these solutions can be written as an integral of the general form

F() =Z

C eR(z)g(z)dz(1.1) whereCis a contour in the complex plane. Moreover, many integral transforms such as the Laplace transform and the Fourier transform have this form. This paper will give a brief introduction to some of the most common methods for nding the asymptotic expansion of such integral representations, with focus on the method of steepest descent and some applications of it. For more detailed information about this topic the text [8] by Peter Miller is especially recommended. See also the following well-known texts: [6] by Carl M. Bender and Steven A. Orszag, [5] by Norman Bleistein and Richard A. Han- delsman, [1] by Mark J. Ablowitz and A. S. Fokas. In Section 2 the denition of asymptotic expansion and some notation will be brought up before going into the part of the methods where the classical result known as Watson's lemma will be presented. In Section 3 we explain Laplace's method, a technique for nding the (dominant) contribution of the asymptotics of a real valued integral coming from a point or points on the interval where

R(t) attains its maximum.

In Section 4 the method of steepest descent is discussed, which can be consid- ered a generalization of Laplace's method used for complex integrals. In Section

5 two applications are presented, the asymptotics of the Airy function and the

asymptotics of the linearized KdV equation, both computed using the method of steepest descent. 3

2Elemen taryasymptotics and W atson's

lemma 2.1

Asymptotic expansion

To begin with there are a couple of denitions and expressions that needs to be brought up to clarify the material in upcoming sections. The asymptotic be- havior of a function is expressed as an asymptotic expansion given a sequence of functions, so to be able to form an asymptotic expansion one needs asymptotic sequences. For the following denitions letDbe a subset of the complex plane. Denition 2.1.(Asymptotic sequence.) A sequence of functionsn(z)1 n=0is called an asymptotic sequence asz!z0from D if whenevern > m, we haven(z) =o(m(z))asz!z0from D.

Example 2.2.The sequence of functionszn1

n=0is an asymptotic sequence asz!0, andzn1 n=0is an asymptotic sequence asz! 1.Denition 2.3.(Asymptotic expansion.) Letn(z)1 n=0be an asymptotic sequence asz!z0from D. Then the sum N X n=0a nn(z) is said to be an asymptotic approximation asz!z0from D of a functionf(z) if f(z)NX n=0a nn(z) =o(N(z)) asz!z0from D. Ifan 1 n=0is a sequence of complex constants such that the above is true for each N, then the formal innite series 1 X n=0a nn(z) is called an asymptotic series and is said to be an asymptotic expansion off(z) asz!z0from D. Normally the asymptotic expansion off(z) is written in the following way: f(z)1X n=0a nn(z) 4 asz!z0. Note that the symbolis used here. That is because the series does not nec- essarily need to converge and the use of an equality sign would therefore be incorrect. Even if the the series is divergent it will still say a lot about the behavior of the functionf(z). Example 2.4.Consider a functionfbeingC1in a neighborhood of the origin. Then from Taylor's theorem we have the asymptotic expansion f(x)1X n=0f (n)(0)n!xn asx!0. Indeed, f(x)NX n=0f (n)(0)n!xn=En(x); where forx2[R;R] the remainder term is E n(x) =f(n+1)(s)(n+ 1)!xn+1 for somesbetween 0 andx. This implies thatEn(x) =O(xn+1), which is stronger thano(xn).Note that a given function may have several asymptotic expansions, and that an asymptotic series does not need to represent a specic function. However, the coecientsanin an asymptotic expansion of a function with respect to a given asymptotic sequence are unique.

Example 2.5.Suppose that

F()1X n=0a n n as! 1and Re >0. If the termeis added toF() the same asymptotic expansion is still valid, i.e. e +F()1X n=0a n n as! 1, with Re >0. We say thateis "beyond all orders" with respect to the asymptotic sequencen1 n=0.5

2.2W atson'slemma

There are many dierent techniques to obtain asymptotic expansions of inte- grals. One elementary method is integration by parts. In this section a partic- ular case of the integral (1.1) will be considered, namely

F() =Z

T 0 et(t)dt:(2.1) This corresponds toR(t) =t, soR(t) has a maximum at the left endpoint of the interval [0;T].

Example 2.6.(Integration by parts.)Suppose

F() =Z

T 0 etg(t)dt; whereg2C1andT2R+. By using integration by parts it follows that the asymptotic expansion of this integral is F()1X n=0g (n)(0) n+1(2.2) as! 1. Indeed, making the rst step with this well-known method one obtains the following:

F() =g(t)et

T 0+Z T 0g

0(t)et

dt:

Continuing repeating this gives

F() =NX

n=0 g(n)(0) n+1g(n)(T)eT n+1 +Z T 0g (n+1)(t)et n+1dt;(2.3)

where the integral in (2.3) isO((N+2)) as! 1. This implies (2.2)Clearly this method is not suited for all types of integrals of the form (1.1).

Another way of nding the asymptotic expansion for integrals is the following classical result known as Watson's lemma, that is a generalization of the exam- ple above. Proposition 2.7.(Watson's lemma.) Suppose thatT2R+and thatg(t)is a complex-valued, absolutely integrable function on[0;T]: Z T 0 jg(t)jdt <1: Suppose further thatg(t)is of the formg(t) =t(t)where >1and(t) has an innite number of derivatives in some neighborhood oft= 0. Then the exponential integral

F() =Z

T 0 etg(t)dt 6 is nite for all >0and it has the asymptotic expansion F()1X n=0 (n)(0)(+n+ 1)n!(+n+1)(2.4) as!+1.

For the proof of this proposition, see [8].

Remark 2.8.In many situationsT= +1. In this case the conditions are unnecessarily restrictive. IfT= +1it suces to assume that(t) =O(eat) as

t! 1for somea2R, and the proposition still holds true.Remark 2.9.From the denition of the gamma function

(z) =Z 1 0 tz1etdt;Rez >0;(2.5) it follows that (z+ 1) =z(z);(2.6) and in particular (n+ 1) =n! forn2N. So in case= 0 the asymptotic expansion expression (2.4) is reduced to F()1X n=0g (n)(0) n+1;

which is the same result as (2.2).Note that the integral expression (2.1) and the Laplace transform off(t) dened

by

L(f)(s) =Z

1 0 estf(t)dt; are of the same form, withT=1. So in this case one can get an asymptotic expansion of a Laplace transform for largesimmediately. For many exponential integrals Watson's lemma can be used by making adjustments of the variables so that the required structure of the lemma is met, and then apply it to obtain a full asymptotic expansion.

Example 2.10.Consider an integral of the form

F() =Z

et2(t)dt;(2.7) where as in Proposition 2:7 the functionis assumed to be innitely dieren- tiable in a neighborhood of 0. Let = min(;), then F()Z et2(t)dt=o(n); 7 for alln2N, as! 1. Because removing the part of the interval that not is around the pointt= 0 will give an error beyond all orders with respect ton1 n=0. Now, using symmetry Z et2(t)dt=Z 0 et2(t)dt+Z 0 et2(t)dt Z 0 et2((t) +(t))dt= 2Z 0 et2even(t)dt;(2.8) where even(t) =((t) +(t))2 Making the substitutiont=ps, the integral (2.8) becomes Z 2 0 eseven(ps)ps ds:(2.9)

By writingeven(t) as its Taylor series we obtain

even(t)12 1X n=0 (n)(0)n!tn+(n)(0)n!(t)n! 12 1 X n=0 (n)(0)n!(1 + (1)n)t2=1X n=0 (2n)(0)(2n)!t2n; which gives us even(ps)1X n=0 (2n)(0)(2n)!sn: The integral (2.9) is now suitable for using Watson's lemma and we observe thatquotesdbs_dbs5.pdfusesText_10